Computing Graph Properties by Randomized Subcube Partitions

  • Ehud Friedgut
  • Jeff Kahn
  • Avi Wigderson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)

Abstract

We prove a new lower bound on the randomized decision tree complexity of monotone graph properties. For a monotone property A of graphs on n vertices, let p = p(A) denote the threshold probability of A, namely the value of p for which a random graph from G(n,p) has property A with probability 1/2. Then the expected number of queries made by any decision tree for A on such a random graph is at least Ω(n2/ max{pn, logn}).

Our lower bound holds in the subcube partition model, which generalizes the decision tree model. The proof combines a simple combinatorial lemma on subcube partitions (which may be of independent interest) with simple graph packing arguments. Our approach motivates the study of packing of “typical” graphs, which may yield better lower bounds.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B78]
    B. Bollobás, Extremal Graph Theory, Academic Press, Chapter 8, 1978.Google Scholar
  2. [C74]
    P. A. Catlin, Subgraphs of graphs I, Discrete Math, 10, pp. 225–233, 1974.Google Scholar
  3. [CKS02]
    A. Chakrabarti, S. Khot and Y. Shi, Evasiveness of subgraph containment and related properties, SIAM J. on Computing, 31, pp. 866–875, 2002.MATHCrossRefMathSciNetGoogle Scholar
  4. [CK01]
    A. Chakrabarti and S. Khot, Improved lower bounds on the randomized complexity of graph properties, Proc. of the 28th ICALP conference, Lecture Notes in Computer Science 2076, pp. 285–296, 2001.Google Scholar
  5. [G92]
    D. H. Gröger, On the randomized complexity of monotone graph properties, Acta Cybernetica, 10, pp. 119–127, 1992.MATHMathSciNetGoogle Scholar
  6. [H91]
    P. Hajnal, An Ω(n4/3) lower bound on the randomized complexity of graph properties, Combinatorica, 11, pp. 131–143, 1991.MATHCrossRefMathSciNetGoogle Scholar
  7. [K88]
    V. King, Lower bounds on the complexity of graph properties, Proc. of the 20th STOC conference, pp. 468–476, 1988.Google Scholar
  8. [KSS84]
    J. Kahn, M. Saks and D. Sturtevant, A topological approach to evasiveness, Combinatorica 4, pp. 297–306, 1984.MATHCrossRefMathSciNetGoogle Scholar
  9. [RV76]
    R. Rivest and J. Vuillemin, On recognizing graph properties from adjacency matrices, TCS 3, pp. 371–384, 1976.CrossRefMathSciNetGoogle Scholar
  10. [SS78]
    N. Sauer and J. Spencer, Edge-disjoint placement of graphs, J. of Combinatorial Theory Ser. B, 25, pp. 295–302, 1978.MATHCrossRefMathSciNetGoogle Scholar
  11. [SW86]
    M. Saks and A. Wigderson, Probabilistic Boolean Decision Trees and the Complexity of Evaluating Game Trees, Proc. of the 27th FOCS conference, pp. 29–38, 1986.Google Scholar
  12. [vN28]
    J. von Neumann, Zur Theorie der Gesellschaftspiele, Matematische An-nalen, 100, pp. 295–320, 1928. English translation On the theory of games and strategy, in “Contributions to the theory of Games”, IV, pp.13-42, 1959.CrossRefMATHGoogle Scholar
  13. [Y87]
    A.C. Yao, Lower bounds to randomized algorithms for graph properties, Proc. of the 28th FOCS conference, pp. 393–400, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Ehud Friedgut
    • 1
  • Jeff Kahn
    • 2
  • Avi Wigderson
    • 3
  1. 1.The Hebrew UniversityIsrael
  2. 2.Rutgers UniversityGermany
  3. 3.Institute for Advanced Study and The Hebrew UniversityIsrael

Personalised recommendations